No, the Azad Hind were a Japanese Ally, so you had Britain + British India vs Japan and Azad Hind. Azad Hind would have had to be fighting both British India and the Japanese to make it a three way war.
The first thing I thought of was the War of the Five Kings in the “Song of Ice and Fire.”
The second thing I thought of was Frankish kingdoms of the middle ages.
They didn’t practice primogeniture; when the father died, his kingdom was divided among his sons. So each son had his own country, and they often fought each other.
Western Sahara - Spain vs. Morocco vs. Mauritania vs. Sahrawis
But this still isn’t a three-way war. The Azad Hind/Indian National Army was a Japanese created, armed, and directed puppet of the Japanese. The Azad Hind was Japanese in the same way that the Indian Army was British. So, still only two sides here.
That was what came to my mind as well. Strictly speaking I’m not sure it fits under the OP’s criteria, since was generally two factions, though not necessarily always the same two factions, allied against the third.
Initially the two smaller factions, Wu and Shu were allied against the much stronger Wei, and won some key battles. When Wu and Shu fell out due to territorial dispute, Shu launched a large invasion against Wu, and it looked at first as if it might succeed. Wei was prepared to invade Wu as well, but the Emperor was persuaded against it by an adviser. After Shu was decisively defeated Wei and Wu launched a joint invasion against it, but the Wei forces were bogged down in the mountain passes and Wu flipped sides again and re-allied with Shu.
I think the Haitian revolt of the early 1800s would probably qualify - the Haitians fought (though not at the same time) against both the British and the French who were also at war with each other. Around 1804-5 (as far as I can determine) the British fleet was supporting the Haitians against the French at one end of the island and the Spanish of Santo Domingo against the Haitians at the other - and this while Britain and Spain were at war.
wasn’t israel technically at war with both iran and iraq in 1979? that was when iraq launched an attack on iran.
He’s not talking about a war with just 3 participants (2 vs 1), he’s talking about a war where:
Country A simultaneously fights countries B and C;
Country B simultaneously fights countries A and C; and
Country C simultaneously fights countries A and B.
I don’t see how WWI or WWII possibly meet that definition. Same with the other examples given thus far.
Even my Western Sahara example?
The first Anglo-Mysore War 1767-1770 involved the Kingdom of Mysore vs British East India Company and allies vs the Maratha Empire and Hyderabad princely state.
Wikipedia doesn’t have a great amount of detail but it sounds like all three sides fought each other at various stages of the conflict.
http://www.onwar.com/aced/data/papa/polisario1975.htm
Polsario fights Spain, Morocco and Mauritania. Mauritania has enough in 1979 and signs peace agreement with Polsario. Morocco moves into military vacuum.
There are military incidents where Moroccans fought Spaniards, but these don’t look like systematic conflict to me. Is this impression wrong?
The US had a cold war against the Soviet Union and China, who had a hot border conflict in 1969. Then Nixon went to China in 1972 and disrupted this promising example.
US -> China via Cambodia
US -> Soviet Union via Vietnam
China -> Soviet Union in 1969 border clash
isn’t that the natural course of maintaining balance in a 3 way war? it wouldn’t be much of a tripartite situation if two sides has an alliance strong enough to systematically devour the third.
implicitly or explicitly, the weaker two sides must whittle away at the strongest party until equilibrium is achieved. what makes it interesting is where the weaker sides appear to call the shots, deciding whether balance should end, or the non-aggression pact.
Two years later, I still say that there was a three-way war between Comanche, Texans/Anglos and Mexicans/Spaniards, each fighting for some level of sovereignty in Texas.
For example, in the years 1840-1841:
Mexico v Comanche: “…In September 1840 and continuing until March 1841 came the first of the great raids. During this period six Comanche armies numbering between two hundred and eight hundred warriors invaded northern Mexico. The most far reaching of the raids reached the Departments of San Luis Potosi and Zacatecas 400 miles south of the Big Bend…”
Comanche v Texas: “The Great Raid of 1840 was the largest raid ever mounted by Native Americans on white cities in what is now the United States… To avenge what the Comanche viewed as a bitter betrayal by the Texans, the Comanche war chief Buffalo Hump raised a huge war party of many of the bands of the Comanche, and raided deep into white-settled areas of Southeast Texas.”
Texas v Mexico: “The Texas Santa Fe Expedition was a commercial and military expedition to secure the Republic of Texas’s claims to parts of Northern New Mexico for Texas in 1841.”
Or one of the weaker parties joins with the stronger realizing they can’t win. In my example thats what eventually happens, there was several Anglo-Mysore wars and in later wars the Maratha Empire joined the british against Mysore, but the first war is counted as a separate war and in that one it was a true three way, Mysore vs British East India Company vs Maratha Empire.
I’m sure if the respective countries all had a “Game Theory Treaty” then yes it would be. Of course if they were really antagonistic to each other, they wouldn’t be sharing intelligence and you’d have to know how much men/material the other country fighting against your enemy (whom you are also fighting against) is expending in order to make an appropriate allocation. So without mutual intelligence sharing (or very good spying, i suppose), there’d be no way to reach an ideal equillibrium. No way to know whether to allocate a bullet against country A or B (if you’re C). Dividing it all up evenly assumes both enemies have equal resources, and present equal imminent dangers. No way to make that presumption intelligently.
But regardless, most people here are answering by talking about wars where 1 country simultaneously fought against another 2, or where 2 countries fought simultaneously against one. I figured I’d spell out exactly what the OP was talking about after so many non-responsive answers.
There’s always the war between Eurasia, Eastasia and Oceania.
BIG BROTHER IS WATCHING YOU
But Oceania was Allied with East Asia against Eurasia. Or were they allied with Eurasia against East Asia? For some reason I always mix that up. 
In the Congo wars of the last fifteen years or so, alliances have frequently shifted, but I believe there have been periods of 3-way hostilities among Congo vs. Rwandan client vs. Angola or Zambia.
Oceania has always been at war with Eastasia.
My best guess, examples are left as exercises for the student, would be during the various “Indian Wars” in US and Canadian history. Two strong sides attack each other across the (recognized by the two sides) territory of a third (weak) group. Third group attacks (futiley (sp!)) both invading armies. I’m not going to bother to look it up, but it almost certainly happened. If you want an example from history, that’s where to look.
Did any of these conflicts have a distinct battle where 3 or more sides all fought each other at the same time and place? I’d like to read about that.